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<b>let </b><font color="Maroon">L</font> be  <a href="waybel_0.html#V24">up-complete</a> <a href="waybel_0.html#NM6">Semilattice</a>;<br/>

<b>assume </b><a NAME="E1:4"/><i><font color="Green">E34</font></i>: 
for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>  ex <font color="Olive">I</font> being  <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font> st <br/>( <font color="Olive">x</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Olive">I</font> &amp; ( for <font color="Olive">J</font> being  <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font>  st <font color="Olive">x</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Olive">J</font> holds <br/><font color="Olive">I</font> <a href="tarski.html#R1">c=</a> <font color="Olive">J</font> ) )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/><b>consider </b><font color="Maroon">I</font> being   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E2:4_1"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">I</font>
 <b>and </b><br/><a NAME="E3:4_1"/><i><font color="Green">E38</font></i>: 
for <font color="Olive">J</font> being  <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font>  st <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Olive">J</font> holds <br/><font color="Maroon">I</font> <a href="tarski.html#R1">c=</a> <font color="Olive">J</font>
 <b>by </b><i/>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:4_1_1"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 ;<br/><b>then reconsider </b><font color="Maroon">y'</font> = <font color="Maroon">y</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> ;<br/><a NAME="E3:4_1_1"/>
for <font color="Olive">J</font> being  <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font>  st <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Olive">J</font> holds <br/><font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Olive">J</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">J</font> be   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font>;<br/>

<b>assume </b><a NAME="E1:4_1_1_1"/>
<font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">J</font>
 ;<br/>

<a NAME="E2:4_1_1_1"/><b>then </b>
<font color="Maroon">I</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">J</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:4_1_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">J</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div><a NAME="E4:4_1_1"/><b>then </b>
<font color="Maroon">y'</font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="waybel_3.html#T21">WAYBEL_3:21</a></i>;<br/><b>hence </b><a NAME="E5:4_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="waybel_3.html#K1">waybelow</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="waybel_3.html#T7">WAYBEL_3:7</a></i>;<br/></div><b>end;</b></div><a NAME="E5:4_1"/><b>then </b><i><font color="Green">E43</font></i>: 
<font color="Maroon">I</font> <a href="tarski.html#R1">c=</a>  <a href="waybel_3.html#K1">waybelow</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:4_1_2"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="waybel_3.html#K1">waybelow</a> <font color="Maroon">x</font>
 ;<br/><b>then reconsider </b><font color="Maroon">y'</font> = <font color="Maroon">y</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> ;<br/><a NAME="E3:4_1_2"/>
<font color="Maroon">y'</font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a> <font color="Maroon">x</font>
 <b>by </b><i>, <a class="ref" href="waybel_3.html#T7">WAYBEL_3:7</a></i>;<br/><b>hence </b><a NAME="E4:4_1_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 <b>by </b><i>, <a class="ref" href="waybel_3.html#T20">WAYBEL_3:20</a></i>;<br/></div><b>end;</b></div><a NAME="E7:4_1"/><b>then </b>
 <a href="waybel_3.html#K1">waybelow</a> <font color="Maroon">x</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">I</font>
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><a NAME="E8:4_1"/><b>then </b>
 <a href="waybel_3.html#K1">waybelow</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">I</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/><b>hence </b><a NAME="E9:4_1"/>
(  <a href="waybel_3.html#K1">waybelow</a> <font color="Maroon">x</font> is   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font> &amp; <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><a href="waybel_3.html#K1">waybelow</a> <font color="Maroon">x</font></span>)</span> &amp; ( for <font color="Olive">I</font> being  <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font>  st <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Olive">I</font> holds <br/> <a href="waybel_3.html#K1">waybelow</a> <font color="Maroon">x</font> <a href="tarski.html#R1">c=</a> <font color="Olive">I</font> ) )
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:4"/>
<font color="Maroon">L</font> is <a href="waybel_3.html#V3">continuous</a>
 <b>by </b><i/>;<br/>


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